Boundary Value Problems In Domains Research Articles

Persistence of regularity of the solution to a hyperbolic boundary value problem in domain with corner

This article is about the question of the persistence of regularity for the solution to hyperbolic boundary value problem in the quarter-space. More precisely we will both consider the pure boundary value problem and the initial boundary value problem and we propose a functional space, based upon the high order Sobolev space in which a control of the data of the problem leads to a control of the solution (in the same space). The space proposed here contains the tangential Sobolev space. The analysis borrows some ideas of the study of characteristic boundary value problems in the half-space for which the good derivative to consider is known to be the tangential derivatives x1∂1 instead of the normal derivative ∂1. For quarter-space problems the good quantity to consider will be the radial derivative x1∂1+x2∂2 and then we recover the control of tangential derivatives x1∂1 and x2∂2 using explicit formulas in polar coordinates. The regularity of the solution is then established intrinsically by adapting the method introduced by the author to deal with half-space problems without using regularization methods.

The regularity problem in domains with lower dimensional boundaries

In the present paper we establish the solvability of the Regularity boundary value problem in domains with lower dimensional boundaries (flat and Lipschitz) for operators whose coefficients exhibit small oscillations analogous to the Dahlberg-Kenig-Pipher condition.The proof follows the classical strategy of showing bounds on the square function and the non-tangential maximal function. The key novelty and difficulty of this setting is the presence of multiple non-tangential derivatives. To solve it, we consider a cylindrical system of derivatives and establish new estimates on the “angular derivatives”.

On Solving Initial Boundary Value Problems for a Parabolic Equation in Dini Classes

The method of potentials is one of the classical methods for solving initial boundary value problems for parabolic equations and systems. The method is based on studying the smoothness of parabolic potentials in various functional spaces and studying the solvability of the corresponding integral equations and systems to which the problems are reduced. This method allows one to find a constructive form of solutions to initial-boundary value problems and directly investigate the smoothness of solutions. This article continues a series of works on constructing the solvability theory of parabolic initial boundary value problems in domains with nonsmooth lateral boundaries in the time variable. The article considers the first and second initial boundary value problems with zero initial condition for a uniformly parabolic equation that is one-dimensional in the spatial variable. It is assumed that the equation coefficients are bounded and uniformly continuous with a modulus of continuity satisfying twice the Dini condition. The specific feature of the study is that the right side of the equation can grow to infinity in a certain way when approaching the initial data assignment line. The solvability (in the classical sense) of these problems is established in a semi-bounded curvilinear domain in a plane with a nonsmooth lateral boundary from the Dini-Hölder class, admitting, in particular, so-called "beaks". The solution is constructed as a sum of the plane parabolic potential and the single layer potential, the kernel of which is the fundamental solution of the equation. The smoothness of the solution of these problems in the Dini-Hölder classes is investigated, and the corresponding correctness estimates are obtained. These estimates characterize the behavior of solutions and their first-order spatial derivative in the closure of the domain. The results obtained can be used to study heat and mass transfer processes. For the proof, the smoothness in the Dini-Hölder space of a plane parabolic potential with an unlimited density of the Dini class in a curvilinear domain with a nonsmooth lateral boundary is investigated. The obtained result can be used in solving other initial boundary value problems for an inhomogeneous parabolic equation.

Laplace Green's functions for infinite ground planes with local roughness

The Green's functions for the Laplace equation satisfying the Dirichlet and Neumann boundary conditions on the upper side of the infinite plane with a circular hole are introduced and studied. These functions enable solutions of the boundary value problems in domains where the hole is closed by an arbitrary mesh (locally rough surfaces). The developed approach accounts for arbitrary positive and negative ground elevations inside the domain of interest, which is not possible to achieve using the regular method of images. Such problems appear in electrostatics, however, the methods developed apply to other domains where the Laplace or Poisson equations govern. Integral and series representations of the Green's functions are provided. Using these Green's functions, an efficient computational technique based on the boundary element method with fast multipole acceleration is developed. A numerical study of some benchmark problems is presented.

Retraction Note: Existence results for the general Schr\xf6dinger equations with a superlinear Neumann boundary value problem

The main goal of this paper is to study the general Schrodinger equations with a superlinear Neumann boundary value problem in domains with conical points on the boundary of the bases. First the formulation and the complex form of the problem for the equations are given, and then the existence result of solutions for the above problem is proved by the complex analytic method and the fixed point index theory, where we absorb the advantages of the methods in recent works and give some improvement and development. Finally, we are also interested in the asymptotic behavior of solutions of the mentioned equation. These results generalize some previous results concerning the asymptotic behavior of solutions of non-delay systems of Schrodinger equations or of delay Schrodinger equations.

On elliptic problems in domains with low dimensional boundaries

We study a solvability for some concrete boundary value problems in domains with a low dimensional boundary. Such boundaries are considered as a limit situation of a volume problem for which one or more parameters of a domain tend to their endpoint values. The solvability conditions in Sobolev-Slobodetskii spaces are given using the wave factorization concept for an alliptic symbol..

Isoperimetric cones and minimal solutions of partial overdetermined problems

In this paper we consider a partial overdetermined mixed boundary value problem in domains inside a cone as in [18]. We show that, in cones having an isoperimetric property, the only domains which admit a solution and which minimize a torsional energy functional are spherical sectors centered at the vertex of the cone. We also show that cones close in the C1,1 -metric to an isoperimetric one are also isoperimetric, generalizing so a result of [1]. This is achieved by using a characterization of constant mean curvature polar graphs in cones which improves a result of [18].

Finite-dimensional approximations to the Poincaré–Steklov operator for general elliptic boundary value problems in domains with cylindrical and periodic exits to infinity

Finite-dimensional approximations to the Poincaré–Steklov operator for general elliptic boundary value problems in domains with cylindrical and periodic exits to infinity

A Nyström method for mixed boundary value problems in domains with corners

In this paper we propose a new approach to the numerical solution of the mixed Dirichlet-Neumann boundary value problem for the Laplace equation in planar domains with piecewise smooth boundaries. We consider a perturbed BIE system associated to the problem and present a Nyström method for its numerical solution. As Mellin type integral operators are involved, we need to modify the method close to the corners in order to prove its stability and convergence. Some numerical tests are also given to show the efficiency of the method here described.

Graded parametric CutFEM and CutIGA for elliptic boundary value problems in domains with corners

We develop a parametric cut finite element method for elliptic boundary value problems with corner singularities where we have weighted control of higher order derivatives of the solution to a neighborhood of a point at the boundary. Our approach is based on identification of a suitable mapping that grades the mesh towards the singularity. In particular, this mapping may be chosen without identifying the opening angle at the corner. We employ cut finite elements together with Nitsche boundary conditions and stabilization in the vicinity of the boundary. We prove that the method is stable and convergent of optimal order in the energy norm and L2 norm. This is achieved by mapping to the reference domain where we employ a structured mesh.

RETRACTED ARTICLE: Existence results for the general Schr\xf6dinger\t\t\tequations with a superlinear Neumann boundary value problem

The main goal of this paper is to study the general Schrödinger\t\t\t\tequations with a superlinear Neumann boundary value problem in domains with conical\t\t\t\tpoints on the boundary of the bases. First the formulation and the complex form of\t\t\t\tthe problem for the equations are given, and then the existence result of solutions\t\t\t\tfor the above problem is proved by the complex analytic method and the fixed point\t\t\t\tindex theory, where we absorb the advantages of the methods in recent works and give\t\t\t\tsome improvement and development. Finally, we are also interested in the asymptotic\t\t\t\tbehavior of solutions of the mentioned equation. These results generalize some\t\t\t\tprevious results concerning the asymptotic behavior of solutions of non-delay\t\t\t\tsystems of Schrödinger equations or of delay Schrödinger equations.

On boundaryvalue problems in domains without (A)-condition

On boundaryvalue problems in domains without (A)-condition

The Exact and Approximate Solutions of Some Boundary Value Problems in Domains with Angular Points of the Boundary

On the basis of the exact solution of biharmonic problems of elasticity theory in a half-strip one possible reason is shown of those problems that arise when an approximate or numerical approaches leading the solution of boundary value problems to infinite systems of linear algebraic equations. Construction of exact solutions of some boundary value problems for differential equations in partial derivatives is not possible without their extensions to Riemann surfaces. Moreover, each of the boundary value problem corresponds to its Riemann surface. This fact is important to consider when developing an effective approximate and numerical methods of solving boundary value problems.

Regularity of solutions to Dirichlet boundary value problem in domains on a manifold

The regularity of solutions to the Dirichlet boundary value problem is studied for elliptic differential operators of order 2m defined on subdomains of a manifold. Relationships between the smoothness of the right hand side, the boundary, and the solutions are obtained.

On the regularity of solutions to variational and boundary-value Problems in Domains with Hölder boundary

On the regularity of solutions to variational and boundary-value Problems in Domains with Hölder boundary

The analytical solution of the 3D model with Robin's boundary conditions for 2 peat layers

<p>In this paper we consider averaging methods for solving the 3-D boundary value problem in domain containing 2 layers of the peat block. We consider the metal concentration in the peat blocks. Using experimental data the mathematical model for calculation of concentration of metal in different points in every peat layer is developed. A specific feature of these problems is that it is necessary to solve the 3-D boundary-value problems for elliptic type partial differential equations of second order with piece-wise diffusion coefficients in every direction and peat layers.</p><p>The special parabolic and exponential spline, which interpolation middle integral values of piece-wise smooth function, are considered. With the help of this splines is reduce the problems of mathematical physics in 3-D with piece-wise coefficients to respect one coordinate to problems for system of equations in 2-D. This procedure allows reduce the 3-D problem to a problem of 2-D and 1-D problems and the solution of the approximated problem is obtained analytically.</p><p>The solution of corresponding averaged 2-D initial-boundary value problem is obtained also numerically, using for approach differential equations the discretization in space applying the central differences. The approximation of the 2-D non-stationary problem is based on the implicit finite-difference and alternating direction (ADI) methods. The numerical solution is compared with the analytical solution.</p>

On the application of the method of difference potentials to linear elastic fracture mechanics

SummaryThe Difference Potential Method (DPM) proved to be a very efficient tool for solving boundary value problems (BVPs) in the case of complex geometries. It allows BVPs to be reduced to a boundary equation without the knowledge of Green's functions. The method has been successfully used for solving very different problems related to the solution of partial differential equations. However, it has mostly been considered in regular (Lipschitz) domains. In the current paper, for the first time, the method has been applied to a problem of linear elastic fracture mechanics. This problem requires solving BVPs in domains containing cracks. For the first time, DPM technology has been combined with the finite element method. Singular enrichment functions, such as those used within the extended finite element formulations, are introduced into the system in order to improve the approximation of the crack tip singularity. Near‐optimal convergence rates are achieved with the application of these enrichment functions. For the DPM, the reduction of the BVP to a boundary equation is based on generalised surface projections. The projection is fully determined by the clear trace. In the current paper, for the first time, the minimal clear trace for such problems has been numerically realised for a domain with a cut. Copyright © 2015 John Wiley & Sons, Ltd.

Cauchy integral and singular integral operator over closed Jordan curves

This paper is mostly a review paper. It contains a description of old and recent results concerning the regularity conditions on a Jordan curve in the plane that imply the boundedness of the singular integral operator as well as the boundary behavior of the Cauchy type integral. These results are of significance for boundary value problems in domains with non-smooth and non-rectifiable boundaries.

Enriched isogeometric analysis of elliptic boundary value problems in domains with cracks and/or corners

SUMMARYThe mapping method was introduced in Jeong et al. (2013) for highly accurate isogeometric analysis (IGA) of elliptic boundary value problems containing singularities. The mapping method is concerned with constructions of novel geometrical mappings by which push‐forwards of B‐splines from the parameter space into the physical space generate singular functions that resemble the singularities. In other words, the pullback of the singularity into the parameter space by the novel geometrical mapping (a non‐uniform rational basis spline (NURBS) surface mapping) becomes highly smooth. One of the merits of IGA is that it uses NURBS functions employed in designs for the finite element analysis. However, push‐forwards of rational NURBS may not be able to generate singular functions. Moreover, the mapping method is effective for neither the k‐refinement nor the h‐refinement. In this paper, highly accurate stress analysis of elastic domains with cracks and ∕ or corners are achieved by enriched IGA, in which push‐forwards of NURBS via the design mapping are combined with push‐forwards of B‐splines via the novel geometrical mapping (the mapping technique). In a similar spirit of X‐FEM (or GFEM), we propose three enrichment approaches: enriched IGA for corners, enriched IGA for cracks, and partition of unity IGA for cracks. Copyright © 2013 John Wiley & Sons, Ltd.

Conformal reduction of boundary problems for harmonic functions in a plane domain with strong singularities on the boundary

We consider the Dirichlet, Neumann and Zaremba problems for harmonic functions in a bounded plane domain with nonsmooth boundary. Purpose: We wish to construct explicit formulas for solutions of these problems when the boundary curve belongs to one of the following three classes: sectorial curves, logarithmic spirals and spirals of power type. Methods: To study the problem, we apply the familiar Vekua-Muskhelishvili method which consists in the use of conformal mapping of the unit disk onto the domain to pull back the problem to a boundary problem for harmonic functions in the disk. This in turn later reduces to a Toeplitz operator equation on the unit circle with symbol-bearing discontinuities of the second kind. Results: We develop a constructive invertibility theory for Toeplitz operators and thus derive solvability conditions as well as explicit formulas for solutions. Conclusions: Our results raise Fredholm theory for boundary value problems in domains with singularities which are not necessarily rectifiable.